1.04a Binomial expansion: (a+b)^n for positive integer n

4

1. Expand \(( 1 + a ) ^ { 5 }\) in ascending powers of \(a\) up to and including the term in \(a ^ { 3 }\).
2. Hence expand \(\left[ 1 + \left( x + x ^ { 2 } \right) \right] ^ { 5 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying your answer.

3

1. Find the first three terms in the expansion of \(( 3 - 2 x ) ^ { 5 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 4 + x ) ^ { 2 } ( 3 - 2 x ) ^ { 5 }\).

4 The coefficient of \(x\) in the expansion of \(\left( 4 x + \frac { 10 } { x } \right) ^ { 3 }\) is \(p\). The coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x + \frac { k } { x ^ { 2 } } \right) ^ { 5 }\) is \(q\). Given that \(p = 6 q\), find the possible values of \(k\).

7

1. Write down the first four terms of the expansion, in ascending powers of \(x\), of \(( a - x ) ^ { 6 }\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + \frac { 2 } { a x } \right) ( a - x ) ^ { 6 }\) is - 20 , find in exact form the possible values of the constant \(a\).

3 The coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 2 x ^ { 2 } + \frac { k ^ { 2 } } { x } \right) ^ { 5 }\) is \(a\). The coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 k x - 1 ) ^ { 4 }\) is \(b\).

1. Find \(a\) and \(b\) in terms of the constant \(k\).
2. Given that \(a + b = 216\), find the possible values of \(k\).

1 The coefficient of \(x ^ { 4 }\) in the expansion of \(( 3 + x ) ^ { 5 }\) is equal to the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 x + \frac { a } { x } \right) ^ { 6 }\). Find the value of the positive constant \(a\).

1 The coefficient of \(x ^ { 3 }\) in the expansion of \(\left( p + \frac { 1 } { p } x \right) ^ { 4 }\) is 144 .
Find the possible values of the constant \(p\).

2

1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 2 + 3 x ) ^ { 4 }\).
2. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 5 }\).
3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + 3 x ) ^ { 4 } ( 1 - 2 x ) ^ { 5 }\).

3

1. Give the complete expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 5 }\).
2. In the expansion of \(\left( a + b x ^ { 2 } \right) \left( x + \frac { 2 } { x } \right) ^ { 5 }\), the coefficient of \(x\) is zero and the coefficient of \(\frac { 1 } { x }\) is 80 . Find the values of the constants \(a\) and \(b\).

1 Find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 2 - 5 x ) ( 1 + 3 x ) ^ { 10 }$$

6 The coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x + \frac { a } { x ^ { 2 } } \right) ^ { 5 }\) is 720 .

1. Find the possible values of the constant \(a\).
2. Hence find the coefficient of \(\frac { 1 } { x ^ { 7 } }\) in the expansion.

1

1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 + x ) ^ { 5 }\).
2. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 6 }\).
3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + x ) ^ { 5 } ( 1 - 2 x ) ^ { 6 }\).

3 Find the term independent of \(x\) in each of the following expansions.

1. \(\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 }\)
2. \(\left( 3 x + \frac { 2 } { x ^ { 2 } } \right) ^ { 6 } \left( 1 - x ^ { 3 } \right)\)

6 In the expansion of \(\left( \frac { x } { a } + \frac { a } { x ^ { 2 } } \right) ^ { 7 }\), it is given that $$\frac { \text { the coefficient of } x ^ { 4 } } { \text { the coefficient of } x } = 3 \text {. }$$ Find the possible values of the constant \(a\).

6 It is given that the coefficient of \(x ^ { 3 }\) in the expansion of $$( 2 + a x ) ^ { 4 } ( 5 - a x )$$ is 432 .
Find the value of the constant \(a\).

5 In the expansion of \(\left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }\), the coefficients of \(x ^ { 6 }\) and \(x ^ { 3 }\) are equal.

1. Find the value of the non-zero constant \(a\).
2. Find the coefficient of \(x ^ { 6 }\) in the expansion of \(\left( 1 - x ^ { 3 } \right) \left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }\).

1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + k x ) ( 1 - 2 x ) ^ { 5 }\) is 20 .
Find the value of the constant \(k\).

5 In the expansion of \(( a + b x ) ^ { 7 }\), where \(a\) and \(b\) are non-zero constants, the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) are the first, second and third terms respectively of a geometric progression. Find the value of \(\frac { a } { b }\).

1

1. Expand \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 }\).
2. Find the first four terms in the expansion, in ascending powers of \(x\), of \(( 1 + 2 x ) ^ { 6 }\).
3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 } ( 1 + 2 x ) ^ { 6 }\).

8

1. It is given that in the expansion of \(( 4 + 2 x ) ( 2 - a x ) ^ { 5 }\), the coefficient of \(x ^ { 2 }\) is - 15 .
   Find the possible values of \(a\).
2. It is given instead that in the expansion of \(( 4 + 2 x ) ( 2 - a x ) ^ { 5 }\), the coefficient of \(x ^ { 2 }\) is \(k\). It is also given that there is only one value of \(a\) which leads to this value of \(k\). Find the values of \(k\) and \(a\).

2

1. Find the first three terms, in ascending powers of \(x\), in the expansion of \(( 1 + a x ) ^ { 6 }\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 3 x ) ( 1 + a x ) ^ { 6 }\) is - 3 , find the possible values of the constant \(a\).

4 The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + \frac { 2 } { p } x \right) ^ { 5 } + ( 1 + p x ) ^ { 6 }\) is 70 .
Find the possible values of the constant \(p\).

3

1. Find the first three terms in ascending powers of \(x\) of the expansion of \(( 1 + 2 x ) ^ { 5 }\).
2. Find the first three terms in ascending powers of \(x\) of the expansion of \(( 1 - 3 x ) ^ { 4 }\).
3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ^ { 5 } ( 1 - 3 x ) ^ { 4 }\).

1

1. Expand \(( 1 + 3 x ) ^ { 6 }\) in ascending powers of \(x\) up to, and including, the term in \(x ^ { 2 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 - 7 x + x ^ { 2 } \right) ( 1 + 3 x ) ^ { 6 }\).

4

1. Expand the following in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
   1. \(( 1 + 2 x ) ^ { 5 }\).
   2. \(( 1 - a x ) ^ { 6 }\), where \(a\) is a constant.
      In the expansion of \(( 1 + 2 x ) ^ { 5 } ( 1 - a x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is - 5 .
2. Find the possible values of \(a\).